Yes, there is a way to write to the event log you are looking for. You don't need to create a new source, just simply use the existent one, which often has the same name as the EventLog's name and also, in some cases like the event log Application, can be accessible without administrative privileges*.
*Other cases, where you cannot access it directly, are the Security EventLog, for example, which is only accessed by the operating system.
I used this code to write directly to the event log Application:
using (EventLog eventLog = new EventLog("Application"))
{
eventLog.Source = "Application";
eventLog.WriteEntry("Log message example", EventLogEntryType.Information, 101, 1);
}
As you can see, the EventLog source is the same as the EventLog's name. The reason of this can be found in Event Sources @ Windows Dev Center (I bolded the part which refers to source name):
Each log in the Eventlog key contains subkeys called event sources. The event source is the name of the software that logs the event. It is often the name of the application or the name of a subcomponent of the application if the application is large. You can add a maximum of 16,384 event sources to the registry.